Method of reducing backscatter through object shaping using the calculus of variations

ABSTRACT

Variational calculus principles are applied directly to the radiation integral to minimize the radar signature of a two- or three-dimensional geometry. In the preferred embodiment, the radiation integral is minimized through the solution to a differential equation generated by Euler&#39;s calculus of variations (CoV) equation. When used in conjunction with a minimizing sequence, the analysis affords a broad search of all possible coefficient values to ultimately arrive at global minima. Compared to existing techniques, the approach locates local extrema quickly and accurately using fewer impedance matrix calculations, and optimization using the invention is possible over a wide band of frequencies and angles. The method is applicable to a wide variety of situations, including the design of stealth platforms.

REFERENCE TO RELATED APPLICATION

This application claims priority from U.S. provisional application Ser.No. 60/152,687, filed Sep. 7, 1999, the entire contents of which areincorporated herein by reference.

FIELD OF THE INVENTION

This invention relates generally to radar signature and, in particular,to a method of reducing radar cross-section and/or echo length.

BACKGROUND OF THE INVENTION

The use of radar is now widespread, for both commercial and militaryuses. In military applications, particularly during times of war, it maybe essential that a vehicle such as an aircraft go undetected.

The radar signature or “cross-section” of an object is a measure of howmuch radar energy is reflected back or “returned” to a source or systemsearching for the object. The greater the signature, the easier it is todetect, track and potentially direct weapon systems against that object.

Radar cross section is also a function of the direction from which anobject is “viewed.” With regard to aircraft, reduced aircraft radarcross-section is most important when viewed from the front, or in the“frontal sector.” Radar cross section is also typically increased in thepresence of externally supported appendages such as weapons, which aretypically mounted on pylons or against the body of the aircraft.

There are several techniques that may be employed to minimize the radarcross section. Broadly, one class of techniques is used to designaircraft having an inherently low radar signature, whereas otherapproaches seek to modify existing aircraft to achieve this samepurpose. Of course, both broad principles may be applied to the samestructure.

As discussed in U.S. Pat. No. 5,717,397, radar cross-section may beminimized using any combination of the following:

1. Shaping the exterior of the aircraft or external features, includingleading/trailing edges, gaps, and seams, such that radar energy isreflected away from potential enemy radars;

2. Aligning leading and trailing edges, gaps and seams at a minimumnumber of similar angles (especially in the top or “plan” view of theaircraft), such that the radar returns from these various features areconcentrated into fewer angles or sectors.

3. Concealing or hiding highly radar reflective aircraft components fromthe “view” of potential enemy radars; and

4. Utilizing materials and coatings in the construction of aircraftcomponents that absorb or diffuse radar energy.

Whether designing a craft for a low radar signature in the first place,or modifying existing craft to achieve a reduced cross. section, theproblem is complex and often mathematically intensive. Cross-sectionoptimization using manual and empirical methods is labor intensive and,although computer methods may be employed to find local minima, globaloptimization is often elusive. Current automated techniques use Z-matrixcalculations, often requiring numerous iterations to achieve dubiousresults. Although such techniques have improved in recent years,existing methods often still require mechanisms to avoid stagnating inlocal minima.

SUMMARY OF THE INVENTION

Broadly, this invention applies variational calculus principles directlyto the radiation integral to minimize radar cross-section and/or echolength. The radiation integral, which is well known to those of skill inantenna design and other disciplines, may be used to determine theelectromagnetic field scattering of a body given the surface current. Inthe preferred embodiment of this invention, the radiation integral isminimized through the solution to a differential equation generated byEuler's calculus of variations (CoV) equation. When used in conjunctionwith a minimizing sequence, the analysis affords a broad search of allpossible coefficient values to ultimately arrive at. global minima.

Compared to existing techniques, the approach locates local extremaquickly and accurately using fewer impedance matrix calculations. Themethod is applicable to a wide variety of situations, including thedesign of stealth platforms. A thorough analysis of the applicabledesign equations is disclosed, which indicate that optimization over awide band of frequencies and angles is possible. Although the examplespresented are in two dimensions, the procedure is readily extensible tothree dimensions.

BRIEF DESCRIPTION OF THE INVENTION

FIG. 1 is a drawing of a geometry used to illustrate the principles ofthe invention;

FIG. 2 is a drawing of a two-dimensional geometry defined by a single,continuous variable;,

FIG. 3 is a block diagram of an iterative optimization process accordingto the invention;

FIGS. 4a through 4 c illustrate the way in which a two-dimensional shapeis iteratively optimized according to the invention;

FIGS. 5a through 5 c depict echo length corresponding to theoptimizations of FIGS. 4a through 4 c;

FIGS. 6a through 6 c illustrate the way in which a two-dimensional shapeis iteratively optimized according to the invention using polar domainprocessing; and

FIGS. 7a through 7 c depict echo length corresponding to theoptimizations of FIGS. 6a through 6 c.

DETAILED DESCRIPTION OF THE INVENTION

This invention exploits Euler's equation to locate local extrema quicklyand accurately. When used in conjunction with a minimizing sequence, theanalysis affords a broad search of all possible coefficient values toultimately arrive at the global minimum.

Introduction to the Design Equations

The analysis begins with the well-known radiation integral in twodimensions [3] given by: $\begin{matrix}{{{E_{z}^{s}\left( \overset{\_}{\rho} \right)} = {\frac{\omega \quad \mu_{0}}{4}{\int_{s}{{J_{z}^{s}\left( {\overset{\_}{\rho}}^{\prime} \right)}{H_{0}^{(2)}\left( {k{{\overset{\_}{\rho} - {\overset{\_}{\rho}}^{\prime}}}} \right)}\quad {l}}}}},\begin{matrix}{{E_{z}^{s}\left( \overset{\_}{\rho} \right)} \equiv \quad {{TM}_{z}\quad {radiated}\quad {field}\quad \left( {m^{- 1}V} \right)}} \\{{J_{z}^{s}\left( {\overset{\_}{\rho}}^{\prime} \right)} \equiv \quad {z - {{directed}\quad {surface}\quad {current}\quad \left( {m^{- 1}A} \right)}}} \\{\omega \equiv \quad {{angular}\quad {frequency}\quad \left( {s^{- 1}{rad}} \right)}} \\{k \equiv \quad {{wave}\quad {number}\quad \left( m^{- 1} \right)}} \\{\mu_{0} \equiv \quad {{permittivity}\quad {of}\quad {free}\quad {space}\quad \left( {m^{- 1}H} \right)}} \\{{\overset{\_}{\rho}}^{\prime} \equiv \quad {{vector}\quad {from}\quad {origin}\quad {to}\quad {geometry}\quad {surface}\quad (m)}} \\{\overset{\_}{\rho} \equiv \quad {{vector}\quad {from}\quad {origin}\quad {to}\quad {observation}\quad {point}\quad (m)}}\end{matrix}} & (1)\end{matrix}$

The geometry in question is depicted in FIG. 1.

Introduction to the Calculus of Variations

If a two-dimensional integral equation can be constructed of the form$\begin{matrix}{{M\lbrack y\rbrack} = {\int_{a}^{b}\quad {F\left( {x,y,{\overset{.}{\left. y \right)}\quad {x}},} \right.}}} & (2)\end{matrix}$

then it can be shown that the differential equation, $\begin{matrix}{{{F_{y} - {\frac{}{x}F_{\overset{.}{y}}}} = 0},} & (3)\end{matrix}$

when solved for y, will ensure that M[y] is a relative extrema. When theequation approaches zero, it ensures that a maximizing or minimizingsolution is being obtained in an optimum sense.

As an example, consider a simplified form of problem that can arise whencharged particles travel in an electromagnetic field or near a linecharge: $\begin{matrix}{{{M\lbrack y\rbrack} = {\int_{x_{1}}^{x_{2}}{\frac{\sqrt{1 + {\overset{.}{y}}^{2}}}{y}\quad {x}}}},} & (4)\end{matrix}$

where M[y] represents energy. Fundamental laws of physics dictate thatcharged particles will seek the path requiring the minimum amount ofenergy to traverse. The problem is to determine what that path may be,and this lends itself directly to a CoV solution. Here, $\begin{matrix}{{F_{y} = {- \frac{\sqrt{1 + {\overset{.}{y}}^{2}}}{y^{2}}}},} & (5) \\{{F_{\overset{.}{y}} = \frac{\overset{.}{y}}{y\sqrt{1 + {\overset{.}{y}}^{2}}}},{{such}\quad {that}}} & (6) \\{{{\frac{}{x}F_{\overset{.}{y}}} = \frac{{\overset{¨}{y}y\sqrt{1 + {\overset{.}{y}}^{2}}} - {\overset{.}{y}\left( {{\overset{.}{y}\sqrt{1 + {\overset{.}{y}}^{2}}} + {y\quad \overset{.}{y}{\overset{¨}{y}/\sqrt{1 + {\overset{.}{y}}^{2}}}}} \right)}}{\left( {y\sqrt{1 + {\overset{.}{y}}^{2}}} \right)^{2}}},{and}} & (7) \\\begin{matrix}{{F_{y} - {\frac{}{x}F_{\overset{.}{y}}}} = \quad {{- \frac{\sqrt{1 + {\overset{.}{y}}^{2}}\left( \sqrt{1 + {\overset{.}{y}}^{2}} \right)^{2}}{{y^{2}\left( \sqrt{1 + {\overset{.}{y}}^{2}} \right)}^{2}}} -}} \\{\quad {\frac{{\overset{¨}{y}y\sqrt{1 + {\overset{.}{y}}^{2}}} - {\overset{.}{y}\left( {{\overset{.}{y}\sqrt{1 + {\overset{.}{y}}^{2}}} + {y\quad \overset{.}{y}{\overset{¨}{y}/\sqrt{1 + {\overset{.}{y}}^{2}}}}} \right)}}{\left( {y\sqrt{1 + {\overset{.}{y}}^{2}}} \right)^{2}}.}} \\{= \quad 0}\end{matrix} & (8)\end{matrix}$

This reduces ultimately to

ÿy+{dot over (y)} ²+1=0.  (9)

Solving for this differential equation produces as a solution,

(x−C ₁)² +y ² =C ₂ ²,  (10)

an offset circular arc. The constants depend on the choices of x₁, x₂,y(x₁) and y(x₂).

Construction of the Design Equations

According to the invention, equation (1) is placed into a formatamenable to the solution of the Euler equation.

Cartesian Format Design Equations

Assume that a two-dimensional geometry can be defined by a singlecontinuous variable, y(x). If this is the case, the entire geometry maybe defined according to curves of FIG. 2. For this geometry, it isassumed that the two curves [y(x), {tilde over (y)}(x)] must meet atsome common point (in this case, B, where both are zero). As a commonpoint for optimization problems involving minimization, this is ajudicious choice because we will assume that the primary minimizationshould occur about θ=0. Note that {tilde over (y)}(x) appears to dependon y(x) in the sense that it is of the opposite sign. Although this isnot required under the invention, for convenience this assumption willbe used in the following calculations.

We begin by recasting the radiation integral of equation (1) into a moreamenable format for the application of the Euler equation:$\begin{matrix}{{{E_{z}^{s}(\rho)} = {{\frac{\omega \quad \mu_{0}}{4}{\int_{A}^{B}{{J_{z}^{s}\left\lbrack \sqrt{x^{2} + {y^{2}(x)}} \right\rbrack}{H_{0}^{(2)}\left( {k\left\lbrack {\rho - {x\quad \cos \quad \theta_{0}} - {{y(x)}\sin \quad \theta_{0}}} \right\rbrack} \right)}\sqrt{1 + {{\overset{.}{y}}^{2}(x)}}\quad {x}}}} + {\frac{\omega \quad \mu_{0}}{4}{\int_{A}^{B}{{{\overset{\sim}{J}}_{z}^{s}\left\lbrack \sqrt{x^{2} + {{\overset{\sim}{y}}^{2}(x)}} \right\rbrack}{H_{0}^{(2)}\left( {k\left\lbrack {\rho - {x\quad \cos \quad \theta_{0}} - {{\overset{\sim}{y}(x)}\sin \quad \theta_{0}}} \right\rbrack} \right)}\sqrt{1 + {\overset{{\overset{.}{\sim}}^{2}}{y}(x)}}\quad {x}}}}}},} & (11)\end{matrix}$

where ${{\overset{.}{y}(x)} = {\frac{}{x}{y(x)}}},$

θ₀ is the angle of observation, ρ→∞, and [A,B] is the range over x onthe surface represented by the symmetric geometry above. From here on,the notation y(x) will be dropped in favor of y.

Using this new equation, now we can begin to consolidate thenomenclature. First, incorporate the large argument approximation forthe Hankel function $\begin{matrix}{{{H_{0}^{(2)}\left( {k\left\lbrack {\rho - X} \right\rbrack} \right)} \cong {\sqrt{\frac{2j}{k\quad {\pi \left\lbrack {\rho - X} \right\rbrack}}}{\exp \left\lbrack {{- j}\quad {k\left\lbrack {\rho - X} \right\rbrack}} \right\rbrack}}},} & (12)\end{matrix}$

for ρ→∞, and X=x cos θ₀+y sin θ₀.

Equation (12) can be further reduced according to $\begin{matrix}{{\sqrt{\frac{2j}{k\quad {\pi \left\lbrack {\rho - X} \right\rbrack}}}{\exp \left\lbrack {{- j}\quad {k\left\lbrack {\rho - X} \right\rbrack}} \right\rbrack}}\overset{\lim}{\rightarrow}{\sqrt{\frac{2j}{k\quad \pi \quad \rho}}{\exp \left\lbrack {{- j}\quad k\quad \rho} \right\rbrack}{{\exp \left\lbrack {j\quad {kX}} \right\rbrack}.}}} & (13)\end{matrix}$

Next, write $\begin{matrix}{\quad {{E_{z}^{s}(\rho)} = {{M\lbrack y\rbrack} = {k_{0}{\int_{s}\quad {F\left( {x,y,{\overset{.}{\left. y \right)}\quad {x}},} \right.}}}}}} & (14)\end{matrix}$

and for convenience write,

F(x,y,{dot over (y)})=J _(z) ^(s)(x,y)A(x)B(x,y)C(x,{dot over (y)}),where  (15)

$k_{0} \equiv {\frac{\omega \quad \mu_{0}}{2}\sqrt{\frac{2j}{k\quad \pi \quad \rho}}{\exp \left\lbrack {{- j}\quad k\quad \rho} \right\rbrack}}$

A(x)≡exp[jkx cos θ₀]

B(x,y)≡exp[jky sin θ₀]${C\left( {x,\overset{.}{y}} \right)} \equiv \sqrt{1 + {\overset{.}{y}}^{2}}$

The optimization can now be more compactly described. Begin by finding

F _(y)=(J _(z) ^(s))_(y) ABC+J _(z) ^(s) AB _(y) C and  (16)

F_({dot over (y)}) =J _(z) ^(x) ABC _({dot over (y)}), such that${\frac{}{x}F_{\overset{.}{y}}} = {{{ABC}_{\overset{.}{y}}\frac{}{x}J_{z}^{s}} + {J_{z}^{s}{BC}_{\overset{.}{y}}\frac{}{x}A} + {J_{z}^{s}A\quad C_{\overset{.}{y}}\frac{}{x}B} + {J_{z}^{s}{AB}\frac{}{x}{C_{\overset{.}{y}}.}}}$

Now calculate, $\begin{matrix}{{\left( J_{z}^{s} \right)_{y} = {\frac{y}{r}\frac{\partial}{\partial r}{J_{z}^{s}(r)}}},{where}} & (17)\end{matrix}$

 r={square root over (x²+y²+L )}, and  (18)

B _(y) =jk sin θ₀ exp[jky sin θ₀ ]=jk sin θ₀ B, such that  (19)

$\begin{matrix}{F_{y} = {\frac{AB}{C}{\left( {{C^{2}\frac{y}{r}\frac{\partial}{\partial r}J_{z}^{s}} + {j\quad k\quad C^{2}\sin \quad \theta_{0}J_{z}^{s}}} \right).}}} & (20)\end{matrix}$

In a similar fashion, it is straightforward to calculate,$\begin{matrix}{{{\frac{}{x}J_{z}^{s}} = {\frac{x + {y\quad \overset{.}{y}}}{r}\frac{\partial}{\partial r}J_{z}^{s}}},} & (21) \\{{{\frac{}{x}A} = {{{jk}\quad \cos \quad \theta_{0}{\exp \left\lbrack {j\quad {kx}\quad \cos \quad \theta_{0}} \right\rbrack}} = {{jk}\quad \cos \quad \theta_{0}A}}},{and}} & (22) \\{{{\frac{}{x}B} = {{{jk}\quad \overset{.}{y}\quad \sin \quad \theta_{0}{\exp \left\lbrack {j\quad {ky}\quad \sin \quad \theta_{0}} \right\rbrack}} = {{jk}\quad \overset{.}{y}\quad \sin \quad \theta_{0}B}}},{and}} & (23) \\{{C_{\overset{.}{y}} = {\frac{\overset{.}{y}}{\sqrt{1 + {\overset{.}{y}}^{2}}} = \frac{\overset{.}{y}}{C}}},{{such}\quad {that}}} & (24) \\{{{{\frac{}{x}C_{\overset{.}{y}}} = \frac{\overset{¨}{y}}{C^{3}}},{and}}{{\frac{}{x}F_{\overset{.}{y}}} = {\frac{AB}{C}{\left( {{\overset{.}{y}\frac{x + {y\quad \overset{.}{y}}}{r}\frac{\partial}{\partial r}J_{z}^{s}} + {{jk}\quad \overset{.}{y}\quad \cos \quad \theta_{0}J_{z}^{s}} + {{jk}\quad {\overset{.}{y}}^{2}\sin \quad \theta_{0}J_{z}^{s}} + {J_{z}^{s}\frac{\overset{¨}{y}}{C^{2}}}} \right).}}}} & (25)\end{matrix}$

At this point, the Euler equation can now be calculated as$\begin{matrix}{{F_{y} - {\frac{}{x}F_{\overset{.}{y}}}} = {\frac{AB}{C}{\left( {{\frac{y - {x\quad \overset{.}{y}}}{r}\frac{\partial}{\partial r}J_{z}^{s}} + {J_{z}^{s}\left\lbrack {\frac{- \overset{¨}{y}}{1 + {\overset{.}{y}}^{2}} + {{jk}\quad \sin \quad \theta_{0}} - {{jk}\quad \overset{.}{y}\quad \cos \quad \theta_{0}}} \right\rbrack}} \right).}}} & (26)\end{matrix}$

Finally, the design equation for minimization reduces to $\begin{matrix}{\left. {{{D\frac{\partial}{\partial r}J_{z}^{s}} + {\left( {{jE} + F} \right)J_{z}^{s}}}}\rightarrow 0 \right.,{{{for}D} = \frac{y - {x\quad \overset{.}{y}}}{r}},{and}} & (27)\end{matrix}$

 E=k[sin θ ₀ −{dot over (y)}cos θ₀], and$F = {\frac{- \overset{¨}{y}}{1 + {\overset{.}{y}}^{2}}.}$

We may want to choose to allow the aft end of the geometry to havefreedom of movement in some cases. To allow this, require [1]

F _({dot over (y)})|_(A) ^(B) =J _(z) ^(s) ABC _({dot over (y)})|_(A)^(B)=0.  (28)

Using previous calculations, this forces the condition

{dot over (y)}(A)={dot over (y)}(B).  (29)

The relevance of these assignments will become apparent shortly.

It should be apparent that the current (J_(z) ^(s)) is not a prioriinformation in the MoM calculation. This implementation thus requiressome form of iteration. The advantage of this technique versustechniques seeking a similar end is that the optimization relationshipis directly between the surface current and shape. As such, optimizationmay be attained without performing costly impedance matrix calculationsfor each iteration, so long as the shape solution does not change soradically as to significantly change the initialization current, thusinvalidating the solution. The iteration thus requires some controlscheme.

Construction of a Minimizing Sequence

Akhiezer (at p. 143) demonstrates a reasonable method devised by V. Ritzfor the construction of a minimizing sequence. This sequence has enjoyedsuccess in a variety of engineering applications [1]. Salient featuresof what are contained in the text are revisited here.

Start again with the functional formula of equation (2), subject to theconditions

y(A)=a ₁ , y(B)=b ₁.  (30)

Assume that the functional argument, ƒ(x,y,{dot over (y)}), iscontinuous in all its arguments and assume further that the function canbe bounded such that

ƒ(x,y,{dot over (y)})≧α|{dot over (y)}| ^(p)+β.  (31)

for α>0, β, p>1. It is shown in Akhiezer [1] that these conditionsguarantee the existence of a minimizing sequence when combined with ajudicious choice of basis functions. Further, and more importantly, thiscondition guarantees a limit on the bounds of the minimizationcoefficients. This is extremely significant since no other RCSminimization approach can guarantee that its results can contain thesolution to a global minimum considering the infinite possiblecombinations of series coefficients.

The series and basis functions are constructed according to thefollowing conditions:

a. φ₀(A)=a₁, φ₀(B)=b₁

b. φ_(k)(A)=φ_(k)(B)=0 (k=1,2,3, . . . )

c. φ₀(x) lies in the region defined by equation (31) with the possibleexception of its endpoints

d. basis function first derivatives are linearly independent

Based on these conditions, the basis functions chosen for this work were$\begin{matrix}{{{{\varphi_{0}(x)} = {a_{1} + {\frac{b_{1} - a_{1}}{B - A}\left( {x - a} \right)}}},{and}}{{\varphi_{k}(x)} = {\left( {x - A} \right)^{k}\left( {x - B} \right)}}{for}{k > 0.}} & (32)\end{matrix}$

Not only does this choice of basis functions satisfy conditions a-dabove, but the condition of equation (29) is satisfied as well (as asimple examination can show). Now if the series coefficients are chosensuch that

y _(n() x)=φ₀(x)+C ₁φ₁(x)+C ₂φ₂(x)+C ₃φ₃(x)+ . . . +C_(n)φ_(n)(x),  (33)

then the original function equation (2) is adequately represented by

M[y _(n)]=Φ(C ₁ ,C ₂ ,C ₃ , . . . C _(n))=Φ({overscore (C)}).  (34)

Since this is the case, we can assume that the only valuable solutionsafter an initial trial, where Φ({overscore (C)})=M, are those thatsubsequently have a solution, Φ({overscore (C)})≦M.

The analysis thus proceeds starting with the reorganization of equation(31) leading to $\begin{matrix}{{{{\int_{A}^{B}{{{{{\overset{.}{\varphi}}_{0}(x)} + {\sum\limits_{i = 1}^{n}{C_{i}{{\overset{.}{\varphi}}_{i}(x)}}}}}^{p}\quad {x}}} \leq \frac{M - {\beta \left( {B - A} \right)}}{\alpha}} = M_{1}},} & (35)\end{matrix}$

and thus $\begin{matrix}{{\left\{ {\int_{A}^{B}{{{\sum\limits_{i = 1}^{n}{C_{i}{{\overset{.}{\varphi}}_{i}(x)}}}}^{p}\quad {x}}} \right\}^{1/p} \leq {M_{1}^{1/p} + \left\{ {\int_{A}^{B}{{{{\overset{.}{\varphi}}_{0}(x)}}^{p}\quad {x}}} \right\}^{1/p}}} = {M_{2}.}} & (36)\end{matrix}$

Now the left side of the above equation can be put into the form$\begin{matrix}{{\sqrt{C_{1}^{2} + C_{2}^{2} + C_{3}^{2} + \ldots + C_{n}^{2}}\left\{ {\int_{A}^{B}{{{\sum\limits_{i = 1}^{n}{K_{i}{{\overset{.}{\varphi}}_{i}(x)}}}}^{p}\quad {x}}} \right\}^{1/p}},{where}} & (37) \\{{K_{i} = \frac{C_{i}}{\sqrt{C_{1}^{2} + C_{2}^{2} + C_{3}^{2} + \ldots + C_{n}^{2}}}},{{and}\quad {{it}'}s\quad {easy}\quad {to}\quad {see}\quad {that}}} & (38) \\{{\sum\limits_{i = 1}^{n}K_{i}^{2}} = 1.} & (39)\end{matrix}$

Because of this final convenient condition, the function $\begin{matrix}\left\{ {\int_{A}^{B}{{{\sum\limits_{i = 1}^{n}{K_{i}{{\overset{.}{\varphi}}_{i}(x)}}}}^{p}\quad {x}}} \right\}^{1/p} & (40)\end{matrix}$

is continuous on the unit circle and, according to a Weierstrasstheorem, assumes a minimum value of 6 on it. All of this leads to thefinal significant condition $\begin{matrix}{\sqrt{C_{1}^{2} + C_{2}^{2} + C_{3}^{2} + \ldots + C_{n}^{2}} \leq {\frac{M_{2}}{\delta}.}} & (41)\end{matrix}$

Ergo, the coefficients used to construct the geometry for RCSminimization have an upper bound on their combined value.

For this work, values of p=2 and β=0 were used for the inequality. Thecoefficient, α, was computed using a Total Least Squares (TLS) techniquecombined with the computation of the functional integrand and {dot over(y)}. This does not guarantee a bound which will only containminimization solutions, but rather approximates that bound.

The advantage of CoV for the analysis of these problems should be clearby this point. Euler's equation offers the ability to locate localextrema quickly and accurately. When combined in this fashion to form aminimizing sequence, the analysis affords a broad search of all possiblecoefficient values to ultimately arrive at the global minimum. Whatremains is to study the effects of coefficient granularity in theapplication of these solutions. Studies so far have not shown that asingle solution will arise out of every iterative approach. Certainly,genetic algorithms and other acceptable search schemes are applicable tothis problem as well. A block diagram of the overall iterative scheme isdepicted in FIG. 3.

Example of Cartesian Domain Processing

The following example shows results from this technique during a typicalrun on a model order of 4. Note the boundary boxes in the figures. Theseare physical constraints placed on the geometry (a “can't be largerthan” box on the outside and a “cannot be smaller than” box on theinside). The derived shape in this example is optimized for a singlefrequency and angle, where the shape is dictated during optimization bythe model order and choice of basis functions. For this particular case,the routine obtains a fairly wide-well solution, but is limited indepth. The theoretical limit (−∞ at a single angle/single frequency) isnot obtained, however. This limit would be more easily approached forhigher model orders. What is particularly interesting about the finaliteration of the shaping approach here is that it does not approach theinner boundary. Often, minimization approaches will tend to iterateclosely to the limiting contour, but in this case additional space isprovided for the boundary box.

Polar Format Design Equations

Assume that a two-dimensional geometry may be defined in the polardomain by r(θ) for θ=0 to θ_(max). Assume further that we desire thegeometry to be symmetric such that {{tilde over (r)}(θ)ε[2π,−θ_(max)]}={r(θ)ε[0,θ_(max)]}, where {tilde over (r)} represents thesymmetric side. With this assumption in mind, we can rewrite (1) as$\begin{matrix}{{{E_{z}^{s}(\rho)} = {\frac{\omega \quad \mu_{0}}{4}{\int_{s}{{J_{z}^{s}\left\lbrack {r(\theta)} \right\rbrack}{H_{0}^{(2)}\left( {k\left\lbrack {\rho - {{r(\theta)}{\cos \left( {\theta - \theta_{0}} \right)}}} \right\rbrack} \right)}\sqrt{{r^{2}(\theta)} + {{\overset{.}{r}}^{2}(\theta)}}\quad {\theta}}}}},} & (42)\end{matrix}$

where ${{\overset{.}{r}(\theta)} = {\frac{}{\theta}{r(\theta)}}},$

θ₀ is the angle of observation, ρ→∞, and S is the surface represented bythe symmetric geometry above. From here on, the notation r(θ) will bedropped for simply r. Now, Euler's equation becomes $\begin{matrix}{{F_{r} - {\frac{}{\theta}F_{\overset{.}{r}}}} = 0.} & (43)\end{matrix}$

First, incorporate the large argument approximation for the Hankelfunction $\begin{matrix}{{H_{0}^{(2)}\left( {k\left\lbrack {\rho - {r\quad {\cos \left( {\theta - \theta_{0}} \right)}}} \right\rbrack} \right)} = {\sqrt{\frac{2j}{k\quad {\pi \left\lbrack {\rho - {r\quad {\cos \left( {\theta - \theta_{0}} \right)}}} \right\rbrack}}}{\exp \left\lbrack {{- j}\quad {k\left\lbrack {\rho - {r\quad {\cos \left( {\theta - \theta_{0}} \right)}}} \right\rbrack}} \right\rbrack}}} & (44)\end{matrix}$

for ρ→∞.

Next, write

$\begin{matrix}{\quad {{E_{z}^{s}(\rho)} = {{M\lbrack r\rbrack} = {k_{0}{\int_{s}\quad {F\left( {\theta,r,{\overset{.}{\left. r \right)}\quad {\theta}},} \right.}}}}}} & (45)\end{matrix}$

and for convenience write,

F(θ,r,{dot over (r)})=J _(z) ^(s) A(θ,r)B(θ,r,{dot over(r)}),where  (46)

$k_{0} \equiv {\frac{\omega \quad \mu_{0}}{4}\sqrt{\frac{2j}{k\quad \pi \quad \rho}}{\exp \left\lbrack {{- j}\quad k\quad \rho} \right\rbrack}}$

A(θ,r)≡exp[jkr cos(θ−θ₀)]${B\left( {\theta,r,\overset{.}{r}} \right)} \equiv \sqrt{r^{2} + {\overset{.}{r}}^{2}}$

The optimization can now be more compactly described. Begin by finding

F _(r)=(J _(z) ^(s))_(r) AB+J _(z) ^(s) A _(r) B+J _(z) ^(s) AB _(r),and  (47)

F_({dot over (r)})=J_(z) ^(s)AB_({dot over (r)}), such that${\frac{}{\theta}F_{\overset{.}{r}}} = {{{AB}_{\overset{.}{r}}\frac{}{\theta}J_{z}^{s}} + {J_{z}^{s}B_{\overset{.}{r}}\frac{}{\theta}A} + {J_{z}^{s}A\frac{}{\theta}{B_{\overset{.}{r}}.}}}$

Now one can calculate,

A _(r) =jk cos(θ−θ₀ )A, and  (48)

$\begin{matrix}{{B_{r} = {\frac{r}{\sqrt{r^{2} + {\overset{.}{r}}^{2}}} = \frac{r}{B}}},{{such}\quad {that}}} & (49) \\{F_{r} = {\frac{A}{B}{\left( {{B^{2}\frac{\partial}{\partial r}J_{z}^{s}} + {{jk}\quad {\cos \left( {\theta - \theta_{0}} \right)}B^{2}J_{z}^{s}} + {rJ}_{z}^{s}} \right).}}} & (50)\end{matrix}$

In a similar fashion, it is straightforward to calculate,$\begin{matrix}{{{\frac{}{\theta}J_{z}^{s}} = {\overset{.}{r}\frac{\partial}{\partial r}J_{z}^{s}}},} & (51) \\{{{\frac{}{\theta}A} = {{{jk}\left\lbrack {{\overset{.}{r}\quad {\cos \left( {\theta - \theta_{0}} \right)}} - {r\quad {\sin \left( {\theta - \theta_{0}} \right)}}} \right\rbrack}A}},{and}} & (52) \\{{B_{\overset{.}{r}} = {\frac{\overset{.}{r}}{\sqrt{r^{2} + {\overset{.}{r}}^{2}}} = \frac{\overset{.}{r}}{B}}},{{such}\quad {that}}} & (53) \\{{{{\frac{}{\theta}B_{\overset{.}{r}}} = \frac{{\overset{¨}{r}\quad r^{2}} - {r\quad {\overset{.}{r}}^{2}}}{B^{3}}},{and}}{\frac{}{\theta}F_{\overset{.}{r}}} = {\frac{A}{B}{\left( {{{\overset{.}{r}}^{2}\frac{\partial}{\partial r}J_{z}^{s}} + {{{jk}\left\lbrack {{r\overset{.}{r}{\cos \left( {\theta - \theta_{0}} \right)}} - {{\overset{.}{r}}^{2}{\sin \left( {\theta - \theta_{0}} \right)}}} \right\rbrack}J_{z}^{s}} + {\frac{{\overset{¨}{r}\quad r^{2}} - {r\quad {\overset{.}{r}}^{2}}}{B^{2}}J_{z}^{s}}} \right).}}} & (54)\end{matrix}$

At this point, the Euler equation can now be calculated as$\begin{matrix}{{F_{r} - {\frac{}{\theta}F_{\overset{.}{r}}}} = {\frac{A}{B}\left( {{r^{2}\frac{\partial}{\partial r}J_{z}^{s}} + {{{jk}\left\lbrack {{\left( {r^{2} + {\overset{.}{r}}^{2} - {r\quad \overset{.}{r}}} \right){\cos \left( {\theta - \theta_{0}} \right)}} + {{\overset{.}{r}}^{2}{\sin \left( {\theta - \theta_{0}} \right)}}} \right\rbrack}J_{z}^{s}} + {\frac{r^{3} + {2r\quad {\overset{.}{r}}^{2}} - {\overset{¨}{r}\quad r^{2}}}{r^{2} + {\overset{.}{r}}^{2}}J_{z}^{s}}} \right)}} & (55)\end{matrix}$

Finally, the design equation for minimization reduces to$\left. {{{r^{2}\frac{\partial}{\partial r}J_{z}^{s}} + {jDJ}_{z}^{s} + {EJ}_{z}^{s}}}\rightarrow 0 \right.,{for}$

 D=k[(r ² +{dot over (r)} ² −r{dot over (r)})cos(θ−θ₀)+{dot over (r)} ²sin(θ−θ₀)], and  (56)

$E = {\frac{r^{3} + {2r\quad {\overset{.}{r}}^{2}} - {\overset{¨}{r}\quad r^{2}}}{r^{2} + {\overset{.}{r}}^{2}}.}$

Example of Polar Domain Processing

The following example shows results from this technique during an idealrun. In general, the polar domain processing approach was far moresensitive in its ability to arrive at a successful result. The derivedshape in this example is optimized for a single frequency and angle,which explains the awkward appearance. Essentially, the routine isattempting to develop competing scatterers on the fore and aft of thetarget thereby causing cancellation. For this successful run, thetheoretical limit of −∞ is approached at 0° (off the fore end of thestructure).

Extensibility of the Technique

In order to apply this technique to three dimensions, a modified versionof the Euler equation may be used in two dimensions. In effect, theEuler equation of equation (3) is expanded according to $\begin{matrix}{{{f_{u} - {\frac{\partial}{\partial x}f_{u_{x}}} - {\frac{\partial}{\partial y}f_{u_{y}}}} = 0},} & (57)\end{matrix}$

for a integral equation defined according to $\begin{matrix}{{{M\lbrack u\rbrack} = {\int_{D}{\int{{f\left( {x,y,u,u_{x},u_{y}} \right)}{x}{y}}}}},} & (58)\end{matrix}$

where x and y are the variates. The method is extensible to an arbitrarynumber of independent variables. The minimization that would arise fromthis equation would be directly analogous to its two dimensionalcounterpart.

It is also desirable to perform the optimization over a broad range ofangles and frequencies in some cases. This is performed by creatinganother set of optimization equations at selected angles and frequenciesof observation. The minimization according to equations (27) and (56) isthen accomplished for each of those selected angles and frequencies(e.g., angles could be every 1° along the well or at 5 strategiclocations throughout the well, frequencies could be similarly chosen).To accomplish this correctly, the user must remember that the surfacecurrent will change with observation angle and frequency as well, ineffect making

J _(z) ^(x) ≡J _(z) ^(x)(r,k,θ ₀).  (59)

Some costing function scheme would have to be applied to cause asuccessful minimization. The total number of equations that would haveto be minimized in a three dimensional optimization would be identicalto the number of equations requiring minimization in two dimensions{(#frequencies)×(#angles)}.

REFERENCES

1. Akheizer, Naum I (translation from the Russian by Aline H. Frink).Calculus of Variations, Blaisdell Publishing Company, New York/London,1962.

2. Balanis, Constantine A. Advanced Engineering Electromagnetics, JohnWiley & Sons, New York, 1989.

3. Skinner, Dr Paul. “AFIT Notes from Course #EE630 Part II”, November1991.

I claim:
 1. A method of shaping a surface for reduced radar signature,comprising the steps of: a) calculating surface current in accordancewith the radiation integral; b) minimizing the radiation integral usingthe calculus of variations; c) determining the shape associated with theminimization; and d) creating a body incorporating the shape.
 2. Themethod of claim 1, wherein steps a) through c) are iterated to achieve amore global minimization.
 3. The method of claim 1, wherein steps a)through c) are carried out in two or three dimensions.
 4. The method ofclaim 1, wherein steps a) through c) are carried out in Cartesian orPolar coordinates.
 5. The method of claim 1, wherein the step ofdetermining the shape associated with the minimization includes thesteps of: e) establishing a parameter space; and f) determiningparametric bounds within the space.
 6. A method of shaping a surface forreduced radar signature, comprising the steps of: a) calculating surfacecurrent in accordance with the radiation integral; b) minimizing theradiation integral using the calculus of variations; c) determining ifthe shape associated with the minimization is properly bounded; and, ifso, repeating steps b) and c) to improve the minimization, and d)creating a body incorporating the shape.
 7. The method of claim 6,wherein: step c) includes establishing a parameter space and parametricbounds; and, if the minimization is not properly bounded, selectingalternative parameters until an acceptable bounding is achieved.
 8. Themethod of claim 6, wherein steps a) through c) are carried out in two orthree dimensions.
 9. The method of claim 6, wherein steps a) through c)are carried out in Cartesian or Polar coordinates.